It is well known that a systematic procedure of obtaining most finite dimensional completely integrable systems is Adler, Kostant and Symes ( AKS) theorem [1]}, [2]} applying to some Lie algebra \(\mathfrak {g}\)  equipped with an ad-invariant non-degenerate bi-linear form. When this scheme is applied to loop algebra and the Fordy-Kulish decomposition scheme is invoked then the NLS [3]} and KdV [4]} equations can be formulated from there. This mechanism can also be applied for the construction of hierarchies too [5]}. In case of the KdV system, the most general construction that can be derived from this AKS procedure is a pair of coupled complex KdV equations [6]}, through construction of the Lax pair:
\(A=Q \quad {\rm and}\quad B=T+[S,Q],\) 
